Kirstein, PT; Kino, GS; (1958) Solution to the Equations of Space Charge Flow by the Method of the Separation of Variables. Journal of Applied Physics , 29 (12) pp. 1758-1767. 10.1063/1.1723041.

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Abstract

The equations for irrotational, electrostatic laminar space‐charge flow (no thermal velocities or normal magnetic field at the cathode) are set up in terms of the action function. The resulting nonlinear partial differential equation is first reduced, by the method of the separation of variables in cylindrical polar coordinates, to the solution of a set of first‐order, nonlinear, ordinary differential equations in one coordinate. It is shown that it is possible to predict axially symmetric, electrostatic, hollow beams from the inside of hollow cylindrical cathodes with space‐charge‐limited emission which asymptote, theoretically, to other extremely thin cylinders—though thermal velocities would, in practice, limit such a convergence. The characteristics of a particular beam, and the electrode system to produce it are shown. The beams which result from the separation of variables in other coordinate systems are described. In spherical polar coordinates one obtains hollow axially symmetric beams from a conical cathode, which may asymptote to other cones; in a less familiar coordinate system, spiral coordinates, sheet beams from equiangular spiral cathodes result. Finally, it is shown how the method may be extended to include magnetic fields transverse to the cathode. Some new solutions to space‐charge flow in crossed electric and magnetic fields from a space‐charge‐limited cathode are mentioned.

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